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Positive geometries and the amplituhedron

Positive geometries are real semialgebraic spaces that are
equipped with a meromorphic ``canonical form" whose residues reflect
the boundary structure of the space. Familiar examples include
polytopes and the positive parts of toric varieties. A central, but
conjectural, example is the amplituhedron of Arkani-Hamed and Trnka.
In this case, the canonical form should essentially be the tree
amplitude of N=4 super Yang-Mills.

I will talk about the definition and examples of positive geometries,
and discuss what is known about the geometry and combinatorics of the
amplituhedron. The talk will be based on various joint works with
Arkani-Hamed, Bai, Galashin, and Karp.